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CSomputer
cience
Foundations
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 1 / 28
COS 140: Foundations of Computer Science
Boolean Algebra
Fall 2018
CSomputer
cience
Foundations
Homework and announcements!
Introduction
Proofs
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 2 / 28
• Reading: Chapter 5
• Homework: exercises 1–6
◦ Exercise 7 for extra credit
◦ Due: 1 week + 1 class from today (i.e., 9/21)
• Don’t forget – recitation this week!
• Slides: online now
CSomputer
cience
Foundations
Problem
Introduction
• Problem
• Boolean algebra
• Operators
• Expressions
• Precedence
Proofs
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 3 / 28
• Computers compute digital logic functions
• Need some way to describe those functions
• For some ordinary mathematical functions, algebra works well
• But the functions we want aren’t numeric, but give true/false (1/0)
values
• What can we use?
CSomputer
cience
Foundations
Boolean algebra
Introduction
• Problem
• Boolean algebra
• Operators
• Expressions
• Precedence
Proofs
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 4 / 28
• Boolean algebra is analogous to “regular” algebra, but for
true/false values
• Well-known, well-developed mathematical foundation for digital
logic.
• Provides a formalism for specifying the functions that we wish to
have performed.
• Provides a mechanism for proving circuits are equivalent.
• Named for George Boole, a 19th-century mathematician.
CSomputer
cience
Foundations
Operators
Introduction
• Problem
• Boolean algebra
• Operators
• Expressions
• Precedence
Proofs
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 5 / 28
Boolean operators correspond to gates and have same truth tables
as corresponding gate.
CSomputer
cience
Foundations
Operators
Introduction
• Problem
• Boolean algebra
• Operators
• Expressions
• Precedence
Proofs
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 5 / 28
Boolean operators correspond to gates and have same truth tables
as corresponding gate.
• NOT: NOT A, ¬A, A, A′
CSomputer
cience
Foundations
Operators
Introduction
• Problem
• Boolean algebra
• Operators
• Expressions
• Precedence
Proofs
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 5 / 28
Boolean operators correspond to gates and have same truth tables
as corresponding gate.
• NOT: NOT A, ¬A, A, A′
• AND: A AND B, A ·B, AB, A ∧B
CSomputer
cience
Foundations
Operators
Introduction
• Problem
• Boolean algebra
• Operators
• Expressions
• Precedence
Proofs
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 5 / 28
Boolean operators correspond to gates and have same truth tables
as corresponding gate.
• NOT: NOT A, ¬A, A, A′
• AND: A AND B, A ·B, AB, A ∧B
• OR: A OR B, A+B, A ∨B
CSomputer
cience
Foundations
Operators
Introduction
• Problem
• Boolean algebra
• Operators
• Expressions
• Precedence
Proofs
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 5 / 28
Boolean operators correspond to gates and have same truth tables
as corresponding gate.
• NOT: NOT A, ¬A, A, A′
• AND: A AND B, A ·B, AB, A ∧B
• OR: A OR B, A+B, A ∨B
• NAND: A NAND B, A|B, A ↑ B, AB
CSomputer
cience
Foundations
Operators
Introduction
• Problem
• Boolean algebra
• Operators
• Expressions
• Precedence
Proofs
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 5 / 28
Boolean operators correspond to gates and have same truth tables
as corresponding gate.
• NOT: NOT A, ¬A, A, A′
• AND: A AND B, A ·B, AB, A ∧B
• OR: A OR B, A+B, A ∨B
• NAND: A NAND B, A|B, A ↑ B, AB
• NOR: A NOR B, A ↓ B, (A+B)
CSomputer
cience
Foundations
Operators
Introduction
• Problem
• Boolean algebra
• Operators
• Expressions
• Precedence
Proofs
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 5 / 28
Boolean operators correspond to gates and have same truth tables
as corresponding gate.
• NOT: NOT A, ¬A, A, A′
• AND: A AND B, A ·B, AB, A ∧B
• OR: A OR B, A+B, A ∨B
• NAND: A NAND B, A|B, A ↑ B, AB
• NOR: A NOR B, A ↓ B, (A+B)• XOR: A XOR B, A⊕B
CSomputer
cience
Foundations
Boolean Algebra Expressions
Introduction
• Problem
• Boolean algebra
• Operators
• Expressions
• Precedence
Proofs
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 6 / 28
• What is an expression?
• Value of an expression ⇐ depends on the values of the variables
• Evaluate expression: assign values to variables, performing the
operations
• Can create an expression for a function by determining when the
function should be 1, then writing an expression that is 1 in only
those cases.
CSomputer
cience
Foundations
Boolean Algebra Expressions
Introduction
• Problem
• Boolean algebra
• Operators
• Expressions
• Precedence
Proofs
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 6 / 28
• What is an expression?
• Value of an expression ⇐ depends on the values of the variables
• Evaluate expression: assign values to variables, performing the
operations
• Can create an expression for a function by determining when the
function should be 1, then writing an expression that is 1 in only
those cases.
Example: Create a 3-variable expression that equals 1 when all of
the inputs are 1, or when one, and only one, input is 0.
CSomputer
cience
Foundations
Boolean Algebra Expressions
Introduction
• Problem
• Boolean algebra
• Operators
• Expressions
• Precedence
Proofs
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 6 / 28
• What is an expression?
• Value of an expression ⇐ depends on the values of the variables
• Evaluate expression: assign values to variables, performing the
operations
• Can create an expression for a function by determining when the
function should be 1, then writing an expression that is 1 in only
those cases.
Example: Create a 3-variable expression that equals 1 when all of
the inputs are 1, or when one, and only one, input is 0.
ABC + ABC + ABC + ABC
CSomputer
cience
Foundations
Operator Precedence for Boolean Algebra
Introduction
• Problem
• Boolean algebra
• Operators
• Expressions
• Precedence
Proofs
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 7 / 28
CSomputer
cience
Foundations
Operator Precedence for Boolean Algebra
Introduction
• Problem
• Boolean algebra
• Operators
• Expressions
• Precedence
Proofs
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 7 / 28
• Subexpressions inside of parentheses, beginning with innermost
parentheses.
• NOT
• AND
• OR
• Evaluate A NAND B as NOT(A ∧B) and A NOR B as
NOT(A ∨B)
• Evaluate subexpressions of equal precedence from left to right.
CSomputer
cience
Foundations
Proofs of Equivalence
Introduction
Proofs
• Equivalence
• Proof by Truth Table
• Algebraic substitution
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 8 / 28
• Two ways: truth tables and algebraic substitution
• Truth table method: if truth table for expression A same as for
expression B, then A ≡ B
• Algebraic substitution method:
◦ Use laws of Boolean algebra to transform one expression into
the other
◦ Proofs have to be convincing to others
◦ Have to provide enough detail to show how one step follows
from another
◦ Have to provide justification for each step
CSomputer
cience
Foundations
Proof by Truth Table
Introduction
Proofs
• Equivalence
• Proof by Truth Table
• Algebraic substitution
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 9 / 28
Best to have column for each input and results of each operator
CSomputer
cience
Foundations
Proof by Truth Table
Introduction
Proofs
• Equivalence
• Proof by Truth Table
• Algebraic substitution
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 9 / 28
Best to have column for each input and results of each operator
Prove that A⊕B is equivalent to (AB) + (AB)
CSomputer
cience
Foundations
Proof by Truth Table
Introduction
Proofs
• Equivalence
• Proof by Truth Table
• Algebraic substitution
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 9 / 28
Best to have column for each input and results of each operator
Prove that A⊕B is equivalent to (AB) + (AB)
A B A⊕B A B AB AB (AB) + (AB)
0 0 0 1 1 0 0 0
0 1 1 1 0 1 0 1
1 0 1 0 1 0 1 1
1 1 0 0 0 0 0 0
CSomputer
cience
Foundations
Proof by Algebraic Substitution
Introduction
Proofs
• Equivalence
• Proof by Truth Table
• Algebraic substitution
Laws
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 10 / 28
• If you are trying to prove that expression 1 is equivalent to
expression 2:
◦ Start with one of the expressions, let’s say 1.
◦ Change it into another expression (say 1′) using an identity
postulate (law).
◦ Continue the process with 1′ until you arrive at 2.
• You must justify every change to the current expression by listing
the identity postulate used.
• Some identity postulates: double negation law, identity law, null
law, idempotent law, and inverse law
CSomputer
cience
Foundations
Double Negation Law
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 11 / 28
• A = A
CSomputer
cience
Foundations
Identity Law
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 12 / 28
• AND form: 1A = A
• OR form: 0 + A = A
CSomputer
cience
Foundations
Null Law
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 13 / 28
• AND form: 0A = 0• OR form: 1 + A = 1
CSomputer
cience
Foundations
Idempotent Law
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 14 / 28
• AND form: AA = A
• OR form: A + A = A
CSomputer
cience
Foundations
Inverse Law
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 15 / 28
• AND form: AA = 0• OR form: A + A = 1
CSomputer
cience
Foundations
Example: Finding Equivalent Circuits
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 16 / 28
Starting with the following expression, find the equivalent expression
that uses the least number of gates (has the smallest number of
boolean operators) and the least number of inputs
(A + A)(BB)
CSomputer
cience
Foundations
Example: Finding Equivalent Circuits
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 16 / 28
Starting with the following expression, find the equivalent expression
that uses the least number of gates (has the smallest number of
boolean operators) and the least number of inputs
(A + A)(BB)
• Don’t know what the final result will be so use algebraic substitution
instead of truth tables
CSomputer
cience
Foundations
Example: Finding Equivalent Circuits
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 16 / 28
Starting with the following expression, find the equivalent expression
that uses the least number of gates (has the smallest number of
boolean operators) and the least number of inputs
(A + A)(BB)
• Don’t know what the final result will be so use algebraic substitution
instead of truth tables
• Follow your intuition about what should be the case and what makes
sense, then justify with a law.
• In other words, have a plan based on what makes sense.
• Sometimes you need to try things to make the plan.
CSomputer
cience
Foundations
Example: Finding Equivalent Circuits
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 17 / 28
(A + A)(BB)
CSomputer
cience
Foundations
Example: Finding Equivalent Circuits
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 17 / 28
(A + A)(BB)
(A + A)(BB) Double Negation Law
CSomputer
cience
Foundations
Example: Finding Equivalent Circuits
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 17 / 28
(A + A)(BB)
(A + A)(BB) Double Negation Law
(A + A)B Idempotent Law
CSomputer
cience
Foundations
Example: Finding Equivalent Circuits
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 17 / 28
(A + A)(BB)
(A + A)(BB) Double Negation Law
(A + A)B Idempotent Law
1B Inverse Law
CSomputer
cience
Foundations
Example: Finding Equivalent Circuits
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 17 / 28
(A + A)(BB)
(A + A)(BB) Double Negation Law
(A + A)B Idempotent Law
1B Inverse Law
B Identity Law
CSomputer
cience
Foundations
Example: Finding Equivalent Circuits
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 17 / 28
(A + A)(BB)
(A + A)(BB) Double Negation Law
(A + A)B Idempotent Law
1B Inverse Law
B Identity Law
A
B
CSomputer
cience
Foundations
Example: Finding Equivalent Circuits
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 17 / 28
(A + A)(BB)
(A + A)(BB) Double Negation Law
(A + A)B Idempotent Law
1B Inverse Law
B Identity Law
A
B
B
CSomputer
cience
Foundations
Commutative Law
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 18 / 28
• AND form: AB = BA
• OR form: A + B = B + A
CSomputer
cience
Foundations
Associative Law
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 19 / 28
• AND form: (AB)C = A(BC)• OR form: (A + B) + C = A + (B + C)
CSomputer
cience
Foundations
Distributive Law
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 20 / 28
• AND form: A(B + C) = AB + AC
CSomputer
cience
Foundations
Distributive Law
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 20 / 28
• AND form: A(B + C) = AB + AC
• OR form: A + BC = (A + B)(A + C)
CSomputer
cience
Foundations
Distributive Law
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 20 / 28
• AND form: A(B + C) = AB + AC
• OR form: A + BC = (A + B)(A + C)• ...or A ∨ (B ∧ C) = (A ∨B) ∧ (A ∨ C)
CSomputer
cience
Foundations
Absorption Law
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 21 / 28
• AND form: A(A + B) = A
• OR form: A + AB = A
CSomputer
cience
Foundations
DeMorgan’s Law
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 22 / 28
• AND form: AB = A + B
• OR form: A+B = A ·B
CSomputer
cience
Foundations
DeMorgan’s Law
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 22 / 28
• AND form: AB = A + B
• OR form: A+B = A ·B
Aside: recall that AB = A NAND B, and A+B = A NOR B
CSomputer
cience
Foundations
DeMorgan’s Law
Introduction
Proofs
Laws
• Double Negation Law
• Identity Law
• Null Law
• Idempotent Law
• Inverse Law
• Example
• Commutative Law
• Associative Law
• Distributive Law
• Absorption Law
• DeMorgan’s Law
Using the Laws
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 22 / 28
• AND form: AB = A + B
• OR form: A+B = A ·B
Aside: recall that AB = A NAND B, and A+B = A NOR B
NOTE:
AB 6= AB
A+B 6= A+B
CSomputer
cience
Foundations
How would you prove DeMorgan’s Law?
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 23 / 28
What approach would you use? Why?
CSomputer
cience
Foundations
Example: Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 24 / 28
Prove AB + AB = (AB)(AB)
CSomputer
cience
Foundations
Example: Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 24 / 28
Prove AB + AB = (AB)(AB)
• (AB)(AB)
CSomputer
cience
Foundations
Example: Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 24 / 28
Prove AB + AB = (AB)(AB)
• (AB)(AB)
• (AB) + (AB) DeMorgan’s Law
CSomputer
cience
Foundations
Example: Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 24 / 28
Prove AB + AB = (AB)(AB)
• (AB)(AB)
• (AB) + (AB) DeMorgan’s Law
• (AB) + (AB) Double Negation
CSomputer
cience
Foundations
Example: Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 24 / 28
Prove AB + AB = (AB)(AB)
• (AB)(AB)
• (AB) + (AB) DeMorgan’s Law
• (AB) + (AB) Double Negation
• (AB) + (AB) Double Negation
CSomputer
cience
Foundations
Example: Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 24 / 28
Prove AB + AB = (AB)(AB)
• (AB)(AB)
• (AB) + (AB) DeMorgan’s Law
• (AB) + (AB) Double Negation
• (AB) + (AB) Double Negation
• AB +AB Def. of parentheses/precedence
CSomputer
cience
Foundations
Example: Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 25 / 28
Prove AB + AB = (AB)(AB)
CSomputer
cience
Foundations
Example: Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 25 / 28
Prove AB + AB = (AB)(AB)
(AB)(AB)
CSomputer
cience
Foundations
Example: Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 25 / 28
Prove AB + AB = (AB)(AB)
(AB)(AB)
XY Let X = (AB), Y = (AB)
CSomputer
cience
Foundations
Example: Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 25 / 28
Prove AB + AB = (AB)(AB)
(AB)(AB)
XY Let X = (AB), Y = (AB)
X + Y DeMorgan’s Law
CSomputer
cience
Foundations
Example: Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 25 / 28
Prove AB + AB = (AB)(AB)
(AB)(AB)
XY Let X = (AB), Y = (AB)
X + Y DeMorgan’s Law
(AB) + (AB) Substitution for X, Y
CSomputer
cience
Foundations
Example: Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 25 / 28
Prove AB + AB = (AB)(AB)
(AB)(AB)
XY Let X = (AB), Y = (AB)
X + Y DeMorgan’s Law
(AB) + (AB) Substitution for X, Y
(AB) + (AB) Double Negation
CSomputer
cience
Foundations
Example: Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 25 / 28
Prove AB + AB = (AB)(AB)
(AB)(AB)
XY Let X = (AB), Y = (AB)
X + Y DeMorgan’s Law
(AB) + (AB) Substitution for X, Y
(AB) + (AB) Double Negation
(AB) + (AB) Double Negation
CSomputer
cience
Foundations
Example: Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 25 / 28
Prove AB + AB = (AB)(AB)
(AB)(AB)
XY Let X = (AB), Y = (AB)
X + Y DeMorgan’s Law
(AB) + (AB) Substitution for X, Y
(AB) + (AB) Double Negation
(AB) + (AB) Double Negation
AB +AB Def. of parentheses/precedence
CSomputer
cience
Foundations
Example: Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 26 / 28
Prove AB + AB = (AB)(AB)
CSomputer
cience
Foundations
Example: Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 26 / 28
Prove AB + AB = (AB)(AB)
• AB + AB
CSomputer
cience
Foundations
Example: Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 26 / 28
Prove AB + AB = (AB)(AB)
• AB + AB
• AB + AB Double Negation
CSomputer
cience
Foundations
Example: Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 26 / 28
Prove AB + AB = (AB)(AB)
• AB + AB
• AB + AB Double Negation
• (AB)(AB) DeMorgan’s Law
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 27 / 28
Prove ACB + C(B + C) = C
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 27 / 28
Prove ACB + C(B + C) = C
• ACB + C(B + C)
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 27 / 28
Prove ACB + C(B + C) = C
• ACB + C(B + C)• ACB + CB + CC Distributive Law
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 27 / 28
Prove ACB + C(B + C) = C
• ACB + C(B + C)• ACB + CB + CC Distributive Law
• ACB + CB + C Idempotent Law
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 27 / 28
Prove ACB + C(B + C) = C
• ACB + C(B + C)• ACB + CB + CC Distributive Law
• ACB + CB + C Idempotent Law
• CB + ACB + C Commutative Law
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 27 / 28
Prove ACB + C(B + C) = C
• ACB + C(B + C)• ACB + CB + CC Distributive Law
• ACB + CB + C Idempotent Law
• CB + ACB + C Commutative Law
• CB + CBA + C Commutative Law
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 27 / 28
Prove ACB + C(B + C) = C
• ACB + C(B + C)• ACB + CB + CC Distributive Law
• ACB + CB + C Idempotent Law
• CB + ACB + C Commutative Law
• CB + CBA + C Commutative Law
• CB + C + CBA Commutative Law
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 27 / 28
Prove ACB + C(B + C) = C
• ACB + C(B + C)• ACB + CB + CC Distributive Law
• ACB + CB + C Idempotent Law
• CB + ACB + C Commutative Law
• CB + CBA + C Commutative Law
• CB + C + CBA Commutative Law
• CB + C Absorption Law
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 27 / 28
Prove ACB + C(B + C) = C
• ACB + C(B + C)• ACB + CB + CC Distributive Law
• ACB + CB + C Idempotent Law
• CB + ACB + C Commutative Law
• CB + CBA + C Commutative Law
• CB + C + CBA Commutative Law
• CB + C Absorption Law
• C + CB Commutative Law
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 27 / 28
Prove ACB + C(B + C) = C
• ACB + C(B + C)• ACB + CB + CC Distributive Law
• ACB + CB + C Idempotent Law
• CB + ACB + C Commutative Law
• CB + CBA + C Commutative Law
• CB + C + CBA Commutative Law
• CB + C Absorption Law
• C + CB Commutative Law
• C Absorption Law
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 28 / 28
Alternate proof of ACB + C(B + C) = C
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 28 / 28
Alternate proof of ACB + C(B + C) = C
• ACB + C(B + C)
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 28 / 28
Alternate proof of ACB + C(B + C) = C
• ACB + C(B + C)• ACB + CB + CC Distributive Law
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 28 / 28
Alternate proof of ACB + C(B + C) = C
• ACB + C(B + C)• ACB + CB + CC Distributive Law
• ACB + CB + C Idempotent Law
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 28 / 28
Alternate proof of ACB + C(B + C) = C
• ACB + C(B + C)• ACB + CB + CC Distributive Law
• ACB + CB + C Idempotent Law
• ABC + CB + 1C Identity Law
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 28 / 28
Alternate proof of ACB + C(B + C) = C
• ACB + C(B + C)• ACB + CB + CC Distributive Law
• ACB + CB + C Idempotent Law
• ABC + CB + 1C Identity Law
• CAB + CB + C1 Commutative Law, twice
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 28 / 28
Alternate proof of ACB + C(B + C) = C
• ACB + C(B + C)• ACB + CB + CC Distributive Law
• ACB + CB + C Idempotent Law
• ABC + CB + 1C Identity Law
• CAB + CB + C1 Commutative Law, twice
• C(AB +B + 1) Distributive Law
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 28 / 28
Alternate proof of ACB + C(B + C) = C
• ACB + C(B + C)• ACB + CB + CC Distributive Law
• ACB + CB + C Idempotent Law
• ABC + CB + 1C Identity Law
• CAB + CB + C1 Commutative Law, twice
• C(AB +B + 1) Distributive Law
• C(1 +AB +B) Commutative Law, twice
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 28 / 28
Alternate proof of ACB + C(B + C) = C
• ACB + C(B + C)• ACB + CB + CC Distributive Law
• ACB + CB + C Idempotent Law
• ABC + CB + 1C Identity Law
• CAB + CB + C1 Commutative Law, twice
• C(AB +B + 1) Distributive Law
• C(1 +AB +B) Commutative Law, twice
• C1 Null Law
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 28 / 28
Alternate proof of ACB + C(B + C) = C
• ACB + C(B + C)• ACB + CB + CC Distributive Law
• ACB + CB + C Idempotent Law
• ABC + CB + 1C Identity Law
• CAB + CB + C1 Commutative Law, twice
• C(AB +B + 1) Distributive Law
• C(1 +AB +B) Commutative Law, twice
• C1 Null Law
• 1C Commutative Law
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 28 / 28
Alternate proof of ACB + C(B + C) = C
• ACB + C(B + C)• ACB + CB + CC Distributive Law
• ACB + CB + C Idempotent Law
• ABC + CB + 1C Identity Law
• CAB + CB + C1 Commutative Law, twice
• C(AB +B + 1) Distributive Law
• C(1 +AB +B) Commutative Law, twice
• C1 Null Law
• 1C Commutative Law
• C Identity Law
CSomputer
cience
Foundations
Example: Another Proof by Algebraic Substitution
Introduction
Proofs
Laws
Using the Laws
• Proving DeMorgan’s
Law
• Proof 1
• Proof 1a
• Proof 1b
• Proof 2
• Proof 3
Copyright c© 2002–2018 UMaine School of Computing and Information Science – 28 / 28
Alternate proof of ACB + C(B + C) = C
• ACB + C(B + C)• ACB + CB + CC Distributive Law
• ACB + CB + C Idempotent Law
• ABC + CB + 1C Identity Law
• CAB + CB + C1 Commutative Law, twice
• C(AB +B + 1) Distributive Law
• C(1 +AB +B) Commutative Law, twice
• C1 Null Law
• 1C Commutative Law
• C Identity Law
⇒ Often more than one way to do proof!
